Maximum-norm estimates for resolvents of elliptic finite element operators

Let Ω Ω be a convex domain with smooth boundary in R d R^d . It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on Ω Ω is analytic with respect to the maximum-norm, uniformly in the mesh-width. This implies a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane, and conversely. Here we show directly that such a resolvent estimate holds outside any sector around the positive real axis, with arbitrarily small angle . This is useful in the study of fully discrete approximations based on A ( θ ) A(θ ) -stable rational functions, with θ θ small.